Paragraph: Given that for each a ∈ ( 0 , 1 ) , lim h → 0 + ∫ h 1 − h t − a ( 1 − t ) a − 1 d t exists. Let this limit be g ( a ) . In addition, it is given that the function g ( a ) is differentia

Question

Paragraph: Given that for each a ∈ ( 0 , 1 ) , lim h → 0 + ​ ∫ h 1 − h ​ t − a ( 1 − t ) a − 1 d t exists. Let this limit be g ( a ) . In addition, it is given that the function g ( a ) is differentiable on ( 0 , 1 ) . Question: The value of g ( 2 1 ​ ) is

TestHub · Accepted Answer

g 1 2 = lim h → 0 + ⁡ ∫ h 1 - h t - 1 2 1 - t - 1 2 d t = ∫ 0 1 d t t - t 2 = ∫ 0 1 d t 1 4 - t - 1 2 2 = sin - 1 ⁡ t - 1 2 1 2 0 1 = sin - 1 ⁡ 1 - sin - 1 ⁡ (-1) = π

Mathematics - Definite Integration Question with Solution | TestHub

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